Question

For \(x ∈ R\), let \(y(x)\) be the solution of the differential equation \((x^2 -5)\)\(\frac{dy}{dx}\)\(-2xy = - 2x\)\((x^2-5)^2\) such that \(y(2) =7\). Then the maximum value of the function \(y(x)\) is

Answer 1

First-Order Separable Differential Equation 

Given:

The differential equation is: \[ (x^2 - 5)\frac{dy}{dx} - 2xy = - 2x(x^2 - 5)^2 \]

Rearranging the equation to solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{2x(x^2 - 5)^2 - 2xy}{x^2 - 5} \]

Now, we have a first-order separable differential equation. Let's separate the variables: \[ \frac{dy}{2x(x^2 - 5)^2 - 2xy} = \frac{dx}{x^2 - 5} \]

Next, we integrate both sides: \[ \int \frac{dy}{2x(x^2 - 5)^2 - 2xy} = \int \frac{dx}{x^2 - 5} \]

Integrating both expressions will give us the function \( y(x) \).